[Paper Review] Numerical Analysis of Automodel Solutions for Superdiffusive Transport

A. B. Kukushkin, V.S. Neverov+2 more|arXiv (Cornell University)|Jan 9, 2018

Granular flow and fluidized beds6 references3 citations

TL;DR

This paper presents a numerical validation of automodel solutions for superdiffusive transport governed by Lévy flights using distributed computing. It demonstrates high accuracy of approximate analytical solutions across a wide range of space-time variables, confirming their reliability for modeling long-tailed step-length distributions in 1D superdiffusive systems.

ABSTRACT

The distributed computing analysis of the accuracy of automodel solutions for the Green's function of a wide class of superdiffusive transport of perturbation on a uniform background is carried out. The approximate automodel solutions have been suggested for the 1D transport equation with a model long-tailed step-length probability distribution function (PDF) with various power-law exponents. These PDFs describe the transport dominated by the Lévy flights. Massive computing experiments were done to verify automodel solutions. The Everest distributed computing platform and the cluster at NRC Kurchatov Institute were used. The results verify the high accuracy of automodel solutions in a wide range of space-time variables and suggest extending the developed method of automodel solutions to a wider class of stochastic phenomena.

Motivation & Objective

• To assess the accuracy of automodel solutions for superdiffusive transport in the presence of long-tailed step-length probability distributions.
• To investigate the validity of approximate analytical solutions under varying power-law exponents in one-dimensional systems.
• To extend the applicability of automodel solutions to a broader class of stochastic transport phenomena.
• To leverage high-performance computing resources to conduct large-scale numerical experiments for robust validation.
• To provide a computational framework for verifying asymptotic solutions in superdiffusive systems with heavy-tailed dynamics.

Proposed method

• Utilized the Everest distributed computing platform and a cluster at the NRC Kurchatov Institute for massive numerical simulations.
• Solved the 1D transport equation with a model long-tailed step-length probability distribution function (PDF) exhibiting power-law behavior.
• Applied automodel solutions—self-similar approximate solutions derived from asymptotic analysis of the Green's function.
• Performed extensive computing experiments across diverse space-time regimes and power-law exponents.
• Compared numerical results with analytical automodel solutions to evaluate accuracy and convergence.
• Employed distributed computing to handle the computational intensity of simulating superdiffusive processes over large domains.

Experimental results

Research questions

• RQ1 How accurate are automodel solutions in approximating the Green's function for superdiffusive transport with long-tailed step-length PDFs?
• RQ2 What is the range of space-time variables over which automodel solutions remain valid for different power-law exponents?
• RQ3 Can the automodel approach be reliably extended to a broader class of stochastic transport phenomena?
• RQ4 How do numerical simulations based on distributed computing validate the theoretical predictions of automodel solutions?
• RQ5 What is the impact of varying power-law exponents on the accuracy of the approximate solutions?

Key findings

• The automodel solutions exhibit high accuracy across a wide range of space-time variables, confirming their robustness.
• Numerical results from distributed computing experiments show strong agreement with analytical automodel solutions for various power-law exponents.
• The method remains accurate even in the asymptotic regime, supporting its use for long-term and large-scale transport modeling.
• The validation supports the extension of the automodel approach to other superdiffusive systems with heavy-tailed distributions.
• The results demonstrate the feasibility of using distributed computing platforms for high-accuracy numerical validation of asymptotic solutions.
• The study confirms that automodel solutions are a reliable approximation tool for superdiffusive transport governed by Lévy flights.